February 14, 2013. We continued with Saxon Math level 4th, lesson 131. We have only 10 more lessons to go. It went much better than I had anticipated. Maybe because some of the topics we had already encountered through different curricula, so they were not exactly new. Math operations on decimals and common fractions still needed some polishing, specially when they required changing from improper fraction to mixed one or vice versa . The need to use a few skills in a sequence confuses Robert. Just changing fractions from one kind to another was not an issue, but doing this as a next step AFTER adding fractions baffled him. Robert also had to be reminded to change cents to dollars before adding money. He knew that he should place a decimal point under a decimal point, but he did not change cents into decimal part of the dollar. He used to do that in the past, but since nobody was practicing that with him for at least 6 months, he forgot.

When I look at the lesson for tomorrow, I noticed, than one of the problem required counting circumference of the circle. That gave me a pause. Many of the previous lessons in Saxon Math dealt with a circle. Robert drew circles using compass. He also drew diameters and radii, and measured their lengths. Finding circumference calls for an introduction of the number Pi. Long ago, very long ago, I had been introducing Pi to 38 typical fifth graders. With the help of the strings we measured the circumferences of the random round objects and divided them by lengths of their diameters. We all came to some approximation of Pi.

Robert can only divide by numbers up to 12. So dividing by lengths of the diameters which might be decimals or multiple digit numbers is not an option.

In a few weeks, I will resume teaching geometry through Firelight textbook/workbook and then we will spend more time on counting circumferences and maybe even areas of the circles. For now, I will stick with a mechanical formula: Circumference = 3.14 times diameter.

Saxon Math, grade 4th is not an easy curriculum to follow. Its approach to teaching/learning is to address simultaneously different topics through small steps and increase their complexity (difficulties) continuously from one lesson to the next. Most of the teachers and parents do not like such methodology. It seems that there is not enough opportunities to practice new skills to reach fluency. To address that I kept making my own pages to help Robert practice new kinds of problems. Before each lesson, I prepare worksheets that address one or two new skills and give Robert an opportunity to practice them before he confronts them on the page of Saxon Math workbook.

There are, however, big advantages to this approach. Not once, when I taught typical students and when I taught Robert, I was confronted by the fact that children who seemed to master one skill through repetitive practice, forgot it as soon as they moved to a different chapter of a textbook/workbook.

For Robert, switching from one kind of problems to another needs to be practiced daily to help with flexibility of thinking and ability to apply his skills in slightly new circumstances. It is a different thing to practice ten times in a row finding the length of a radius as a half of the length of a diameter than to, for instance, be required to draw a circle knowing its diameter AFTER solving a few unrelated arithmetic problems, counting total price, or reading a graph. While completing a page of similar problems helps with fluency, and thus cannot be avoided, such completion, even errorless, cannot be considered a criterion for mastering the skill. It is the application of the skill in a new context, new situation, and without the support of similar examples that attests to a solid learning.

I use the fact that each unit in Saxon Math (Grade 1 to 4) has two parts/pages: A and B. Problems in part B match problems from part A requiring the student to use similar technique. For a student who, like Robert, is very reluctant to independently solve problems that don’t rely on mechanical skills (algorithms) , having an opportunity to look at the problem #3 on page A to solve problem #3 on page B, is a very important step toward controlling his own learning processes. I don’t mind that Robert looks for clues. I not only encourage but strongly suggest to go back and read the previous solution and then use similar tactic. Looking back is an important tool to have as it allows Robert to loosen his dependence on clues coming from me or his teachers. This is not an independence yet, but it is a half a step toward it.